GRE Example Problem – Order of Operations

Learn more about what you need to know to do well on GRE Math by taking some time to complete this GRE Math example problem.

If L = (a – b) – c and R = a – (b – c), then L – R = ?

This example problem is an exercise in basic mathematical principles, particularly the Order of Operations and your understanding of the Commutative, Associative, and Distributive Laws of mathematics. Let’s do a brief review:

The Commutative Law essentially states that, when you add or multiply, you can swap the order of numbers and get the same answer. So, for example:

Addition: X + Y = Y + X

Multiplication: A x B = B x A

The Associative Law states pretty much the same as the Commutative Law with the additional declaration that when you are multiplying and adding groups of numbers, the grouping of those numbers is irrelevant. So, for example:

Addition:(X + Y) + Z = (Z + Y) + X

Multiplication: (A x B) x C = (C x B) x A

The Distributive Law says you get the same answer when you multiply a number by a group of numbers added together or multiply each number separately and then add them together. So, for example:

A x (B + C) = AB + AC

This might be easier to understand with actual numbers:

3 x (4 +5) = 3(4) + 3(5)

3 x 9 = 12 + 15

27 = 27

The Order of Operations determines the order in which certain mathematical operations act. The actual order of operations is Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction. A particularly useful mnemonic device to remembering this (rather than memorizing the acronym PEMDAS) is “Please Excuse My Dear Aunt Sally.”

Okay, now duly armed, let’s return to the question above:

L – R = [(a – b) – c] – [a – (b – c)]

Notice that each equation has been bracketed off from the other. This is not because you cannot add or subtract these equations; it is only to signify and help you recognize that you are, in fact, beginning with and looking at the two different variables, L and R. Mainly, in this problem, brackets will help you keep track of which numbers are positive and negative.

In order to solve this problem, the first thing you should do is distribute the negative in front of the equation R represents, a – (b – c). The reason for this is that this equation includes two subtractions; so, when you subtract R from L, you will inevitably subtract a negative. Subtracting a negative turns that negative into a positive number. Observe:

L – R = [(a – b) – c] – [a – (b – c)]

L – R = [(a – b) – c] – [a – b + c]

L – R = [(a – b) – c] – a + b – c

After having successfully distributed the negative, the Commutative and Associative Laws, and the Order of Operations, tells us that we are free to solve this problem with no more hang ups: